12Root Locus
With 
let an open‐loop transfer function 
be given by
(12.1)
Consider a proportional feedback controller as given in Figure 12.1.
Figure 12.1. System block diagram.
The closed‐loop transfer function is
In this simple proportional feedback control system we are interested in the movement of the closed‐loop poles as the value of 
is varied. Specifically, as 
varies from 0 to 
we want to determine the location of the roots of
If one desires the root locations for 
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